
This drawing is at a scale of 1 pixel = 10km. The earth's radius here is 636
pixels and the height of the satellite 85 pixels. The scan angle phi is 54
degrees. The drawing is not really precise but accurate enough as a
visualization of the Satellite/Earth/Scanner relationship.

These are the symbols used in this drawing:

R    = Radius of Earth = 6357km
H    = Altitude of Satellite = 850km
phi  = Scanner angle
beta = The angle subtended by the interception point of phi at the Earth's
       surface and the center of Earth.
x    = The distance between the interception point above and a line from the 
       center of Earth to the Satellite.
y    = The distance of the above line from the sub satellite point.

To calculate beta we start with the following relationship:

 x = R * sin(beta) = (H + y) * tan(phi)  (1)
 
 y = R - R * cos(beta) = R * (1 - cos(beta))   (2)

 R * sin(beta) = (H + R * (1 - cos(beta))) * tan(phi)
 
 R * sin(beta) - H * tan(phi) - R * tan(phi) + R * cos(beta) * tan(phi) = 0 (3)

 Dividing (3) by R and re-arranging:

 sin(beta) - H / R * tan(phi) - tan(phi) + cos(beta) * tan(phi) = 0

 sin(beta) + cos(beta) * tan(phi) - ( 1 + H / R ) * tan(phi) = 0  (4)

 To solve for beta we use the Newton-Raphson numerical method:
 Xn+1 = Xn - f(X) / f'(X)

 Here we have:

 f(X)  = sin(beta) + cos(beta) * tan(phi) - ( 1 + H / R ) * tan(phi)
 f'(X) = cos(beta) - sin(beta) * tan(phi)
 beta(n+1) = beta(n) - f(beta) / f'(beta)  (5)

 This algorithm is used to calculate beta for a given value of phi.
 The new number of pixels in the rectified image is calculated from:
 N = n * (R * beta_max) / (H * phi_max)  (6)
 where phi_max = 0.9425 rad (54 degrees) and beta_max the corresponding value.

 This is because the satellite's scanner effectively projects the ever-longer
 segments along the arc on the earth's surface to pixels of constant size
 along an arc of radius H.

 DISCLAIMER! Last time I performed any calculations like the above, was
 during my college years, early 1970's!! So please view the above with
 caution!

